# [seqfan] Re: A surprising property of sums of binomial(n,i)

israel at math.ubc.ca israel at math.ubc.ca
Tue Jun 2 01:25:50 CEST 2015

```I think all that's happening here is that row n has n sums, most of which
are near 2^(n/2). The probability of a number of this size being prime is
about 1/log(2^(n/2)) = const/n. So essentially we have n "nearly
independent" events, each with probability approximately const/n. Using a
Poisson approximation, the probability that none of the n occur would
approach a constant. The value of this constant shouldn't be taken too
seriously, because the events aren't really independent: in particular
there are correlations having to do with divisiblility by small primes. But
it does suggest that A258483(n)/n might approach a nonzero constant.

Cheers,
Robert

On Jun 1 2015, Vladimir Shevelev wrote:

>Dear Seqfans,
>
>Before submission sequences A258126, A258483, I naturally
>believed that with the growth of n, the frequency of appearance
>at least one prime of the form sum{0<=i<=k} biomial(n,i) , k=2,...,n-1,
>increases.
>However, studying the Peter's b-file for A258126 I noticed that
>this frequency slowly decreases, and correspondingly the freaquency
>of appearance of terms of A258483 slowly increases. I asked
>Peter to write a table of change of the freaquency with step 100.
>He get
>
>The first few counts of terms of A258483 in steps of 100s
>
>
> {22,56,89,131,170,209,253,292,337,373,420,469,511,566,597,642,687,734,783,823,860,918,963,1017,1065,1122,1172,1214,1266,1313,1365,1417,1461,1511,1562,1597,1649,1694,1737,1770,1797,1860,1918,1974,2025,2079,2133,2191,2255,2305}
>
>Ratios:
>0.22
>0.28
>0.296667
>0.3275
>0.34
>0.348333
>0.361429
>0.365
>0.374444
>0.373
>0.381818
>0.390833
>0.393077
>0.404286
>0.398
>0.40125
>0.404118
>0.407778
>0.412105
>0.4115
>0.409524
>0.417273
>0.418696
>0.42375
>0.426
>0.431538
>0.434074
>0.433571
>0.436552
>0.437667
>0.440323
>0.442813
>0.442727
>0.444412
>0.446286
>0.443611
>0.445676
>0.445789
>0.445385
>0.4425
>0.438293
>0.442857
>0.446047
>0.448636
>0.45
>0.451957
>0.45383
>0.456458
>0.460204
>0.461
>
> In whole the growth continues. Since the growth rate decreases, it
> indicates either to the limit or to the maximum (or small oscillations
> near a number). In any case, it is a surprising property of rows of
> binomials which I still cannot explain. If anyone can?
>
>Best regards,